Cosmological exact solutions of Petrov type D of a nonlinear asymptotic fluid to a dark energy fluid in real and complex geometries with double singularity. Second case.

Abstract In this paper, exact solutions to the Einstein’s equations are obtained for an anisotropic and homogeneous symmetry of Petrov Type D from a nonlinear fluid that responds to the equation of state Q+Q− = 0 where Q± = √ μ1 (−2P1 + μ1) ± 2 BP1 a wherein μ1 = μ − Λ, P1 = P + Λ, and μ, P and Λ are the volumetric energy density, the pressure, and a constant linked to the concept of dark energy. That equation of state and what it represents in certain limits of time (when t→ 0 and when t → ∞) are also analyzed. Two general solutions which are different because of the degree of initial expansion that a coordinate can have in relation to a perpendicular plane are obtained. For each solution, two cases are present: one represents a space-time with real geometry (R) for all the values of t, and asymptotically in time, this case becomes an isotropic space-time of FLRW of a dark energy fluid; and the other one presents a double singularity, so that since the first singularity, space-time is complex (C) until a certain time t = a ( when the second singularity is present) from which space-time is real (R) and with the increase of time, it tends to an isotropic space-time of FLRW from a dark energy fluid. Then, temperature behavior in relation to time is obtained.